We study a local version of the Minority Game, where agents are placed on the nodes of a directed graph. Agents care about being in the minority of the group of agents they are currently linked to and employ myopic best-reply rules to choose their next-period state. We show that, in this benchmark case, the smaller the size of local networks, the larger long-run population-average payoffs. We then explore the collective behavior of the system when agents can: (i) assign weights to each link they hold and modify them over time in response to payoff signals; (ii) delete badly-performing links (i.e., opponents) and replace them with randomly chosen ones. Simulations suggest that, when agents are allowed to weigh links but cannot delete/replace them, the system self-organizes into networked clusters that attain very high payoff values. These clustered configurations are not stable and can easily be disrupted, generating huge subsequent payoff drops. If, however, agents can (and are sufficiently willing to) discard badly performing connections, the system quickly converges to stable states where all agents get the highest payoff, independently of the size of the networks initially in place.